A382848 a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k)^2 * binomial(n+k,k).
1, 1, -5, -35, -29, 751, 3991, -4115, -137885, -495269, 2114245, 25786795, 50109775, -627370925, -4643568305, -495798035, 157753390435, 768269873875, -1851203127335, -35924154988865, -107001450483779, 763444753890721, 7510024190977105, 8899910747771995
Offset: 0
Keywords
Programs
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Mathematica
Table[Sum[(-1)^(n - k) Binomial[n, k]^2 Binomial[n + k, k], {k, 0, n}], {n, 0, 23}] Table[(-1)^n HypergeometricPFQ[{-n, -n, n + 1}, {1, 1}, -1], {n, 0, 23}] Table[SeriesCoefficient[1/(1 + x + x y + y z + x z + x y z), {x, 0, n}, {y, 0, n}, {z, 0, n}], {n, 0, 23}]
Formula
(59*n-94)*n^2*a(n) = 5*(59*n^3-153*n^2+117*n-30)*a(n-1) - (2301*n^3-8268*n^2+9257*n-3050)*a(n-2) - 2*(59*n-35)*(n-2)^2*a(n-3) with a(0) = 1, a(1) = 1 and a(2) = -5. - Peter Bala, May 24 2025
Comments